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Airy functions and transition between semiclassical and harmonic oscillator approximations for one-dimensional bound states
Theoretical and Mathematical Physics ( IF 1.0 ) Pub Date : 2020-08-01 , DOI: 10.1134/s0040577920080024
A. Yu. Anikin , S. Yu. Dobrokhotov , A. V. Tsvetkova

We consider the one-dimensional Schrodinger operator with a semiclassical small parameter $$h$$ . We show that the “global” asymptotic form of its bound states in terms of the Airy function “works” not only for excited states $$n\sim1/h$$ but also for semi-excited states $$n\sim1/h^\alpha$$ , $$\alpha>0$$ , and, moreover, $$n$$ starts at $$n=2$$ or even $$n=1$$ in examples. We also prove that the closeness of such an asymptotic form to the eigenfunction of the harmonic oscillator approximation.

中文翻译:

一维束缚态的半经典和谐振子近似之间的艾里函数和转换

我们考虑具有半经典小参数 $$h$$ 的一维薛定谔算子。我们表明,根据艾里函数的束缚态的“全局”渐近形式不仅适用于激发态 $$n\sim1/h$$,而且适用于半激发态 $$n\sim1/h ^\alpha$$ , $$\alpha>0$$ ,此外,在示例中, $$n$$ 开始于 $$n=2$$ 甚至 $$n=1$$ 。我们还证明了这种渐近形式与谐振子近似的特征函数的接近性。
更新日期:2020-08-01
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